Cauchy Principal ValueDate: 05/23/2000 at 05:03:43 From: Derek Waldron Subject: Cauchy Principal Value Hi Dr. Math, Can you tell me exactly what the Cauchy Principal Value is? I know it's used when a singularity lies on the contour, but all the literature I've read doesn't actually DEFINE what the value is. Thanks, Derek Waldron Undergraduate Physics Student Date: 05/23/2000 at 10:07:40 From: Doctor Rob Subject: Re: Cauchy Principle Value Thanks for writing to Ask Dr. Math, Derek. The Cauchy Principal Value of a divergent integral is the limit as the radius goes to zero of the integral of the function outside a circular neighborhood of the singularity. The simple examples are Riemann integrals along the real line. For example, the integral of y = x + 1/x from -1 to 2 diverges because there is a singularity at x = 0. Then the CPV of the integral is the limit of the integral outside the interval (-r,r), as r -> 0: -r 2 CPV = lim INTEGRAL (x+1/x) dx + INTEGRAL (x+1/x) dx r->0 -1 r -r 2 = lim [x^2 + ln(|x|)] + [x^2 + ln(|x|)] r->0 x=-1 x=r = lim r^2 + ln(r) - 1 - 0 + 4 + ln(2) - r^2 - ln(r) r->0 = lim 3 + ln(2) r->0 = 3 + ln(2) If we took, instead, the interval (-r,2*r), we would get a different limit. This is an artifact of the divergence of the integral. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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